This post is authored by Eran Nevo. (It is the first in a series of five posts.)
What are the possible face numbers of triangulations of spheres?
There is only one zero-dimensional sphere and it consists of 2 points.
The one-dimensional triangulations of spheres are simple cycles, having vertices and edges, where .
The 2-spheres with vertices have edges and triangles, and can be any integer . This follows from Euler formula.
For higher-dimensional spheres the number of vertices doesn’t determine all the face numbers, and for spheres of dimension a characterization of the possible face numbers is only conjectured! This problem has interesting relations to other mathematical fields, as we shall see. Continue reading